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Three Dimensional Geometry

Question
CBSEENMA12035714
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Find the value of p, so that the lines:
are perpendicular to each other. Also find the equations of a line passing through a point (3, 2, -4) and parallel to line l1.

Solution

The equation of line L1:
fraction numerator 1 minus straight x over denominator 3 end fraction equals fraction numerator 7 straight y minus 14 over denominator straight p end fraction equals fraction numerator straight z minus 3 over denominator 2 end fraction rightwards double arrow space fraction numerator straight x minus 1 over denominator negative 3 end fraction space equals fraction numerator straight y minus 2 over denominator begin display style straight p over 7 end style end fraction space equals fraction numerator straight z minus 3 over denominator 2 end fraction space space... left parenthesis 1 right parenthesis The space equation space of space line space straight L subscript 2 colon fraction numerator 7 minus 7 straight x over denominator 3 straight p end fraction equals fraction numerator straight y minus 5 over denominator 1 end fraction equals fraction numerator 6 minus straight z over denominator 5 end fraction rightwards double arrow fraction numerator straight x minus 1 over denominator begin display style fraction numerator negative 3 straight p over denominator 7 end fraction end style end fraction equals fraction numerator straight y minus 5 over denominator 1 end fraction equals fraction numerator straight z minus 6 over denominator negative 5 end fraction... left parenthesis 2 right parenthesis
 Since space line space straight L subscript 1 space and space straight L subscript 2 space are space perpendicular space to space each space other comma space we space have minus 3 cross times open parentheses fraction numerator negative 3 straight p over denominator 7 end fraction close parentheses plus straight p over 7 cross times 1 cross times 2 cross times left parenthesis negative 5 right parenthesis space equals space 0 rightwards double arrow fraction numerator 9 straight p over denominator 7 end fraction plus straight p over 7 equals 10 rightwards double arrow 10 straight p space equals 70 rightwards double arrow straight p equals space 7 Thus space equations space of space lines space straight L subscript 1 space and space straight L subscript 2 space are colon fraction numerator straight x minus 1 over denominator negative 3 end fraction equals fraction numerator straight y minus 2 over denominator 1 end fraction equals fraction numerator straight z minus 3 over denominator 2 end fraction fraction numerator straight x minus 1 over denominator negative 3 end fraction equals fraction numerator straight y minus 5 over denominator 1 end fraction equals fraction numerator straight z minus 6 over denominator negative 5 end fraction
Thus the equation of the line passing through the point (3, 2, -4) and parallel to the line  L1 is:
fraction numerator straight x minus 3 over denominator negative 3 end fraction space equals space fraction numerator straight y minus 2 over denominator 1 end fraction space equals space fraction numerator straight z plus 4 over denominator 2 end fraction 

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